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Properties

Label 2-72-9.7-c3-0-5
Degree $2$
Conductor $72$
Sign $-0.253 + 0.967i$
Analytic cond. $4.24813$
Root an. cond. $2.06110$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.78 + 4.38i)3-s + (−8.65 − 14.9i)5-s + (1.18 − 2.05i)7-s + (−11.5 − 24.4i)9-s + (26.1 − 45.2i)11-s + (−6.84 − 11.8i)13-s + (89.9 + 3.70i)15-s − 82.9·17-s − 126.·19-s + (5.72 + 10.9i)21-s + (27.1 + 47.0i)23-s + (−87.4 + 151. i)25-s + (139. + 17.2i)27-s + (106. − 184. i)29-s + (112. + 194. i)31-s + ⋯
L(s)  = 1  + (−0.535 + 0.844i)3-s + (−0.774 − 1.34i)5-s + (0.0640 − 0.110i)7-s + (−0.427 − 0.904i)9-s + (0.715 − 1.23i)11-s + (−0.146 − 0.253i)13-s + (1.54 + 0.0636i)15-s − 1.18·17-s − 1.53·19-s + (0.0594 + 0.113i)21-s + (0.246 + 0.426i)23-s + (−0.699 + 1.21i)25-s + (0.992 + 0.123i)27-s + (0.681 − 1.18i)29-s + (0.649 + 1.12i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.253 + 0.967i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.253 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $-0.253 + 0.967i$
Analytic conductor: \(4.24813\)
Root analytic conductor: \(2.06110\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{72} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 72,\ (\ :3/2),\ -0.253 + 0.967i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.416486 - 0.539974i\)
\(L(\frac12)\) \(\approx\) \(0.416486 - 0.539974i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (2.78 - 4.38i)T \)
good5 \( 1 + (8.65 + 14.9i)T + (-62.5 + 108. i)T^{2} \)
7 \( 1 + (-1.18 + 2.05i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (-26.1 + 45.2i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (6.84 + 11.8i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + 82.9T + 4.91e3T^{2} \)
19 \( 1 + 126.T + 6.85e3T^{2} \)
23 \( 1 + (-27.1 - 47.0i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-106. + 184. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-112. - 194. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 32.2T + 5.06e4T^{2} \)
41 \( 1 + (250. + 433. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-6.41 + 11.1i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (104. - 181. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 371.T + 1.48e5T^{2} \)
59 \( 1 + (2.90 + 5.03i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-302. + 523. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (377. + 653. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 43.4T + 3.57e5T^{2} \)
73 \( 1 - 671.T + 3.89e5T^{2} \)
79 \( 1 + (-324. + 562. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (67.2 - 116. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 206.T + 7.04e5T^{2} \)
97 \( 1 + (-726. + 1.25e3i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.79046279110824336740199264041, −12.48547093477995770349339206081, −11.59919085558911877592863510127, −10.62252492073193778467606160344, −9.010583880172123994242527988560, −8.434815063939118023077311059299, −6.33911369973064737489732563184, −4.87175120988279087593451480516, −3.87815165367553907896411203179, −0.46735021420415633227617212473, 2.31182166758298249459667846443, 4.40554722920985747657343631650, 6.60528768023941667007322682854, 6.97884433557927590712820022617, 8.432314732670337572744040575712, 10.31317045437170488974332898503, 11.30129440937769803144658808139, 12.09353166230473628395622834386, 13.24622707243852200909950482399, 14.66475365449898672695810052762

Graph of the $Z$-function along the critical line